Homogeneous Pinning Systems: A Class of Exactly Solved Models

摘要

We introduce a class of statistical mechanics non-disordered models -the homogeneous pinning models - starting with the particular case of random walk pinning. We solve the model in the sense that we compute the precise asymptotic behavior of the partition function of the model. In particular, we obtain a formula for the free energy and show that the model exhibits a phase transition, in fact a localization/delocalization transition.We focus in particular on the critical behavior, that is on the behavior of the system close to the phase transition. The approach is then generalized to a general class ofMarkov chain pinning,which is more naturally introduced in terms of (discrete) renewal processes. We complete the chapter by introducing the crucial notion of correlation length and by giving an overview of the applications of pinning models. Ising models are presented at this stage because pinning systems appear naturally as limits of two dimensional Ising models with suitably chosen interaction potentials. In spite of the fact that these lecture notes may be read focusing exclusively on pinning, the physical literature on disordered systems and Ising models cannot be easily disentangled. So a full appreciation of some physical arguments/discussions in these notes does require being acquainted with Ising models.